Fully complementary higher dimensional partitions

We introduce a symmetry class for higher dimensional partitions - fully complementary higher dimensional partitions (FCPs) - and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely quasi transpose complementary plane partitions are equinumerous to symmetric plane partitions.


Introduction
A plane partition π is an array (π i,j ) of non-negative integers with all but finitely many entries equal to 0, which is weakly decreasing along rows and columns, i.e., π i,j ≥ π i+1,j and π i,j ≥ π i,j+1 ; see Figure 1 (left) for an example.MacMahon [12] introduced them at the end of the 19th century as two dimensional generalisations of ordinary partitions and proved in [14] two enumeration results: He showed that the generating function of plane partitions is given by where the sum is over all plane partitions and |π| is defined as the sum of the entries of π.
A plane partition π is said to be contained in an (a, b, c)-box if the entries of π are at most c and π i,j = 0 implies i ≤ a and j ≤ b.The plane partition in Figure 1 is contained in a (3,4,4)-box or any box of larger size.MacMahon showed that the weighted enumeration of plane partitions inside an (a, b, c)-box is given by where the sum is over all plane partitions contained in an (a, b, c)-box.
While plane partitions were already introduced at the end of the 19th century, they came into the focus of the combinatorics community mainly in the second half of the last century.One major goal was to prove the enumeration formulas for the ten symmetry classes of plane partitions.These classes are defined via combinations of the three operations reflection, rotation and complementation which are best defined by viewing plane partitions as lozenge tilings: First, we represent a plane partition π as stacks of unit cubes by placing π i,j unit cubes at the position (i, j), see Figure 1 (middle).By further displaying the shape of the (a, b, c)-box in which we regard the plane partition in and forgetting the shading of the cubes, we obtain a lozenge tiling of a hexagon with side lengths a, b, c, a, b, c, see Figure 1 (right).Interestingly, this was first observed by David and Tomei [7] in 1989.The operation reflection is defined as vertical reflection of the lozenge tiling, rotation as rotation by 120 degrees and completion as rotation by 180 degree.While MacMahon [13] already considered plane partitions invariant under reflection, the operation completion was first described by Mills, Robbins and Rumsey [15] in 1986.A systematic study of the ten symmetry classes which are defined through these operations was initiated by Stanley [18,19] and finished in 2011 by Koutschan, Kauers and Zeilberger [9].For a more detailed overview see [11].
Already in 1916, MacMahon [14] introduced a further generalisation of partitions to arbitrary dimension, namely higher-dimensional partitions.A d-dimensional partition π is an array (π i 1 ,...,i d ) of non-negative integers with all but finitely many entries equal to 0, such that π i ≥ π i+e k for all indices i = (i 1 , . . ., i d ) and 1 ≤ k ≤ d, where e k denotes the k-th unit vector.We say that π is contained in an (n 1 , . . ., n d+1 )-box if all entries are at most n d+1 and π i > 0 implies that i j ≤ n j for all 1 ≤ j ≤ d.Contrary to dimension 1 (partitions) and dimension 2 (plane partitions), there are hardly any results known for higher-dimensional partitions in dimension 3 or higher.MacMahon conjectured a generating formula for each dimension d but it was disproved by Atkin, Bratley, Macdonald and McKay [2] in 1967; see also [8].Only recently the first enumeration result for higher-dimensional partitions was presented by Amanov and Yeliussizov [1].They were able to "correct" MacMahons formula and showed that where the sum is over all d-dimensional partitions, and cor and | • | ch are certain statistics defined in [1,Section 4 and 5].
In this paper we introduce a new symmetry class for plane partitions, namely quarter complementary plane partitions (QCPPs), which can be generalised immediately to higherdimensional partitions.Instead of presenting the definition for QCPPs (it follows from the corresponding definition for higher-dimensional partitions in Section 2.2) we aim to convey the geometric intuition of this symmetry class next.
Let π = (π i,j ) be a plane partition inside an (a, b, c)-box and define by π ′ = (π a+1−i,b+1−j ) i,j the "dual partition" of π inside the (a, b, c)-box.Geometrically, we think of the dual partition as stacks of unit cubes hanging from the ceiling of the (a, b, c)-box instead of standing at its floor, where the first stack is positioned at the corner with coordinates (a, b, c) instead of the corner with coordinates (1, 1, 1), see Figure 2 (left) for a sketch.It is not difficult to see, compare for example with [11,Section 6], that π is self-complementary if π and π ′ fill the (a, b, c)-box without overlap when regarded as stacks of unit cubes.In the array perspective this means π i,j + π ′ i,j = c for all 1 ≤ i ≤ a and 1 ≤ j ≤ b.We can now generalise this idea.Let C be a set of corners of the (a, b, c)box.We define a partition π to be C-complementary if we can fill the (a, b, c)-box without overlap by copies of π placed at the corners of C similar to before, i.e., if a copy of π is placed at a corner of the form ( * , * , c) then its stacks of unit cubes hang from the ceiling If |C| = 2 and the two corners are on a common face of the box, it is not difficult to see that π is determined along the direction orthogonal to this face.Up to rotation there is only one configuration of the two corners such that they do not share a common face: C = {(1, 1, 1), (a, b, c)}.The C-complementary plane partition in this case are selfcomplementary plane partitions.Similarly, if two corners in C are the endpoints of an edge of the box, then π is determined along the direction of this edge.The only configuration of four corners for which no pair of them are endpoints of an edge is up to rotation C = {(1, 1, 1), (a, b, 1), (1, b, c), (a, 1, c)}, see Figure 2 (right) for a sketch.We call the C-complementary plane partitions in this case quarter complementary plane partitions.
In Section 2 we generalise this geometric approach to higher dimensions by using higher dimensional Ferrers diagrams.Translating the obtained criteria on Ferrers diagrams back to the array description for higher-dimensional partitions, we obtain in Lemma 2.1 the definition of d-dimensional fully complementary partitions (FCPs).In Proposition 2.2 we describe the recursive structure of FCPs which implies immediately our main result.
In Section 3 we consider the "typical" symmetry classes for plane partitions restricted to 2-dimensional FCPs, i.e. quarter complementary plane partitions.It turns out that there exists at most one symmetric QCPP in an (a, b, c)-box and that a QCPP can be neither cyclically symmetric nor transpose-complementary.By introducing two variations, namely quasi symmetric and quasi transpose-complementary, we are able to show enumerative results for the corresponding symmetry classes and combinations thereof for QCPPs, see Proposition 3.1 to Proposition 3.4.It is an immediate question if the enumeration of plane partitions under these variations of symmetry classes have closed expressions.In Section 4 we consider these new symmetry classes and similar generalisations.We present three conjectures, the corresponding data generated in computer experiments is given in Appendix A, and the proof of the next result.Theorem 1.2.A plane partition π inside an (n, n, c)-box is called quasi transposecomplementary if π i,j + π n+1−j,n+1−i = c holds for all 1 ≤ i, j ≤ n with i = n + 1 − j.The number of quasi transpose-complementary plane partitions inside an (n, n, c)-box is equal to the number of symmetric plane partitions inside an (n, n, c)-box.
We present different proofs of the above theorem and how it is related to known results on plane partitions, lozenge tilings and perfect matchings respectively.For odd c we relate quasi transpose-complementary plane partitions to transpose-complementary plane partitions and obtain immediately the following numerical connection between symmetric and transpose-complementary plane partitions which seems to be new where TCPP(n, n, 2c) denotes the number of transpose-complementary plane partitions inside an (n, n, 2c)-box and SPP(n, n, c) denotes the number of symmetric plane partitions inside an (n, n, c)-box.Since the proof of the above theorem is computational, it is still an open question if one can find a bijection between symmetric plane partitions and quasi transpose-complementary plane partitions.

Fully complementary Ferrers
where the order relation is component-wise the order relation of the integers.For a positive integer n, we define [n] = {1, . . ., n}.We say that λ is contained in an (n 1 , . . ., n d+1 )-box for positive integers n is a bijection between d-dimensional Ferrers diagrams and d-dimensional partitions which respects the property of being contained in an (n 1 , . . ., n d+1 )-box.Therefore, we identify for the remainder of this paper a d-dimensional Ferrers diagram with the d-dimensional partition it is mapped to.
This can be seen easily by calculating the according images under ρ I for even sized I which are given by There are three further Ferrers diagrams which are fully complementary inside (2, 2, 2, 2), namely In the above definition of fully complementary we restricted ourselves to boxes with even side lengths.The definition can be extended to any side lengths, however, as we see next, either there are no such Ferrers diagrams, or we obtain a set of Ferrers diagram which is in bijection to one where the box has only even side lengths.First assume that there exist k < l such that the k-th and l-th side lengths of the box are given by 2n k + 1 and 2n l + 1 respectively and that λ is a Ferrers diagram which is fully complementary inside this box.Let us regard the point P = (1, . . ., 1, n k , 1, . . ., 1, n l , 1, . . ., 1) whose components are 1 except for the k-th and l-th component.Then there exists an I ⊂ [d + 1] of even size such that P ∈ ρ I, n (λ), where n = ( n 1 , . . ., n d+1 ) and n i is half of the side length of the box in the direction of i-th standard vector.Let I ′ be the set I ′ = (I \ {k, l}) ∪ ({k, l} \ I).It is immediate that |I ′ | is even and that P ∈ ρ I ′ , n (λ) which is a contradiction.Hence there exists no fully complementary Ferrers diagram inside a box with at least two odd side lengths.Now let all side lengths of the box be even with the exception of one side length.Without loss of generality we can assume that the box is a (2n 1 , . . ., 2n d , 2n d+1 + 1)-box.For a fully complementary partition π in this box, we see that a point of the form (i 1 , . . ., i d , n d+1 + 1) is exactly in Again we omit the subscript n and write γ I whenever n is clear from context.The next lemma rephrases the conditions of being fully complementary directly for a d-dimensional partition.
Lemma 2.1.Let n 1 , . . ., n d+1 be positive integers.A d-dimensional partition π is fully complementary inside a (2n 1 , . . ., 2n d+1 )-box if and only if ) for all non-empty subsets J ⊆ [d] of even size and for all (i 1 , . . . Proof.Let λ be a fully complementary Ferrers diagram inside a (2n Denote by FCP(n 1 , . . ., n d+1 ) the set of fully complementary partitions inside a (2n 1 , . . ., 2n d+1 )-box.For 1 ≤ k ≤ d we define the map1 ϕ k : FCP(n 1 , . . ., n d+1 ) → FCP(n 1 , . . ., n k + 1, . . ., n d+1 ) as and the map ϕ d+1 : FCP(n 1 , . . ., n d+1 ) → FCP(n 1 , . . ., n d , n d+1 + 1) as It is not difficult to see, that these maps are well-defined.Let π be the fully complementary partition inside the (4, 4, 4)-box displayed next.As we see in a moment, it is useful to extend the definition of fully complementary partitions to "empty boxes".In particular we define FCP(n 1 , . . ., n d+1 ) to consist of the "empty array" in case that one n k 0 is equal to 0 and all other n i are positive.We extend the map ϕ k to these sets, where ϕ k is the identity (mapping the empty array onto the empty array) if k = k 0 and mapping the empty array to The maps ϕ k allow us to prove the following recursive structure for FCPs.
Proposition 2.2.Let n = (n 1 , . . ., n d+1 ) be a sequence of positive integers.Then FCP(n) is equal to the disjoint union where e k is the k-th unit vector.
Proof.First we prove that the images of the ϕ k are disjoint.Let k < l be integers with ϕ k FCP(n−e k ) ∩ϕ l FCP(n−e l ) = ∅ and let π be an element of this intersection.First, we assume that l = d + 1.Since π ∈ ϕ k (n − e k ) we have by definition π n 1 ,...,n d = n d+1 .On the other hand, π ∈ ϕ d+1 FCP(n − e d+1 ) implies π n 1 ,...,n d ≥ n d+1 + 1 which is a contradiction.Now let l < d + 1.By definition of ϕ k , ϕ l we have This implies ) .Hence let us assume that π n = n d+1 .By (2.1), γ I (π) n = 0 for all I ⊂ [d] of even size.By applying any γ I for an odd sized I ⊆ [d] to (2.1), we see that there is at most one I of odd size for which γ I (π) n is non-zero.Hence by (2.2) there exists exactly one I ⊆ [d] with π n + γ I (π) n = 2n d+1 which implies γ I (π) n = π ρ I (n) = n d+1 .Let k ∈ I and define n = (1, . . ., n k , . . ., 1) as the vector whose entries are 1 except on position k, where the entry is n k and i = (i 1 , . . ., i d ) as the vector with i k = n k and i j ≤ n j for all 1 ≤ j = k ≤ d.Since π is a d-dimensional partition, we have the following inequalities.
Since π ρ {k} (n) = π n+e k > 0 and there exists only one non-empty I with π ρ I (n) = 0, this implies that I = {k} and hence π n+e k = n d+1 .Furthermore we have π n + π n+e k ≥ π n + π n+e k = 2n d+1 by the above inequalities and π n + π n+e k ≤ 2n d+1 by (2.2) since γ {k} (π) n = π n+e k .This implies π n = π n+e k = n d+1 and hence π i = π i+e k = n d+1 for all i defined as before.Denote for l = k by n l the vector with all components equal to 1 except of the k-th and n-th component which are n k or n l respectively.Since ρ {k,l} ( n l ) = n l + e k + e l and ρ {k,l} ( n l + e k ) = n l + e l it follows from (2.1) that π n+e l = π n+e l +e k = 0 and hence π i 1 ,...,i d = 0 for all (i 1 , . . ., i d ) such that i k ∈ {n k , n k + 1} and there exists an i j > n j .Denote by π ′ the array obtained by deleting all entries for which the i-th component of the index is either n k or n k + 1.It is not difficult to verify that π ′ ∈ FCP(n − e k ) and π = ϕ k (π ′ ) which proves the claim.
As we see next, Theorem 1.1 is now a direct consequence of the above proposition.
Proof of Theorem 1.1.Let us denote by Z(x) the left hand side of (1.4), i.e., Z(x) = where the sum is over all n ∈ N d+1 where the j-th component is 0 and all the other components are positive.It is immediate that since each of the FCP(n) in the sum has exactly one element, namely the "empty array".Using Proposition 2.2, we rewrite Z(x) as By bringing all Z(x) terms on one side and using the explicit formula for Z j (x) we obtain the assertion.

Symmetry classes of QCPPs
3.1.(Quasi) symmetric QCPPs.Remember that a plane partition π is quarter complementary inside a (2a, 2b, 2c)-box if for all 1 ≤ i ≤ 2a and 1 ≤ j ≤ 2b we have and for π i,j > 0 exactly one of the following equations holds: Regarded as a regular plane partition, π is called symmetric if a = b and π i,j = π j,i for all 1 ≤ i, j ≤ 2a.By (3.1) we see that π can only be symmetric if the entries on its anti-diagonal are 0, i.e., π i,2a+1−i = 0 for 1 ≤ i ≤ 2a.Together with (3.2) this implies π a,a = 2c and hence that the only symmetric quarter complementary plane partition is .
In order to obtain more interesting objects we omit the symmetry condition on the antidiagonal and call a quarter complementary plane partition π quasi symmetric if π i,j = π j,i for all 1 ≤ i, j ≤ 2a and i = 2a+1−j.Denote by QS(a, c) the set of quarter complementary plane partitions inside a (2a, 2a, 2c)-box which are quasi symmetric.
By defining QS(a, c) to consist of the "empty array" if either a or c is equal to 0, we obtain immediately the following corollary.
Corollary 3.2.The generating function for quasi symmetric quarter complementary plane partitions is given by 3.2.Cyclically symmetric QCPPs.A plane partition π is called cyclically symmetric if a point (i, j, k) in its Ferrers diagram implies that (j, k, i) is also in its Ferrers diagram.As we see next, there is no cyclically symmetric quarter complementary plane partition.Let π be a cyclically symmetric quarter complementary Ferrers diagram inside a (2a, 2a, 2a)box.It is not difficult to see, that being quarter complementary implies that one of the points (1, 1, 2a), (1, 2a, 1) or (2a, 1, 1) has to be part of π, and hence all of them since π is cyclically symmetric.The set ρ {1,2} (π) therefore contains the points (2a, 1, 1) and (1, 2a, 1) which is a contradiction to π ∩ ρ {1,2} (π) = ∅.Contrary to the previous subsection, we did not find an "interesting" generalisation of cyclically symmetric to "quasi cyclically symmetric" for which we can deduce either a result or a conjecture in the case of QCPPs.

Quasi symmetry classes of plane partitions
In the previous section we introduced variations of two symmetry classes for quarter complementary plane partitions.The aim of this section is to consider these and similar symmetry classes for plane partitions.The following three conjectures were found by computer experiments.We have added the according data as well as explicit guessed enumeration formulas for small values of one of the parameters in the appendix.where p a (c) is an irreducible polynomial in Q[c] that is even, i.e., p a (c) = p a (−c).Further the common denominator of the coefficients of p a (c) is a product of "small primes".
Conjecture 4.2.We call a plane partition π inside an (a, a, c)-box quasi transpose complementary of second kind (QTC2), if π is transpose complementary except along the diagonal, i.e., π i,j + π a+1−j,a+1−i = c for all 1 ≤ i, j ≤ a with i = j.Then for a ≥ 2, the number qtcpp 2 (a, c) of QTC2 plane partitions inside an (a, a, c)-box is given by where p a (c) is an irreducible polynomial in Q[c] that is even.Further the common denominator of the coefficients of p a (c) is a product of "small primes".
where p a (c) is an irreducible polynomial in Q[c] that is even.Further the common denominator of the coefficients of p a (c) is a product of "small primes".
Denote by d( π) the number of anti-diagonal entries which are equal to c = c 2 and define the weight ω( π) as Since |{π n−j+1,j , c − π n−j+1,j }| = 1 implies that c is even and π n−j+1,j = c, it is clear that there are ω( π) many QTCPPs mapping to π by the above map.Hence the number of QTCPPs is equal to the weighted enumeration of QTCPPs π whose anti-diagonal entries are at least c 2 .In the following we present two (and a half) proofs for the weighted enumeration of these π.
For odd c, each π corresponds to a plane partition with entries at most c for which the i-th row from top has at most n + 1 − i positiv entries.The number of these plane partitions can be found in [16,Corollary 4.1], compare also with [4,Corollary 4.1].For even c, each π corresponds to a lozenge tiling in the "top half" of an hexagon with side lengths n, n, c, n, n, c, where the bottom of the region is a zig-zag shape directly below the centre line of the hexagon, and each lozenge at the bottom of the region is weighted by 1  2 , see Figure 3 (right) for an example.The number of these tilings is given in [4,Corollary 4.3].The assertion follows in both cases by using the explicit formulas from [4,16].
Denote by qtcpp(n, n, c) the number of QTCPPs inside an (n, n, c)-box and denote by M (R) the number of perfect matchings of a region R. We can rephrase the above observations as qtcpp(n, n, 2c) = 2 n M (P ′ n,n,c ), qtcpp(n, n, 2c + 1) = 2 n M (P n,n,c ), where the Regions P n,n,c is defined in [4, Section 4, Fig. 5] and P ′ n,n,c is defined in [4, Section 4, Fig. 10].On the one side we have by Ciucus factorization theorem for graphs with reflective symmetry [3] the identity PP(a, a, 2c) = 2 n M (P ′ n,n,c )M (P n−1,n−1,c ), where PP(n, n, 2c) denotes the number of plane partitions inside an (n, n, 2c)-box (compare for example with [3] or [4,Equation (4.16)]).This implies the assertion for even c if we know it for odd c and vice versa by using the explicit formulas for the number of (symmetric) plane partitions.Further it is well known, that 2 n M (P ′ n,n,c ) = SP P (n, n, 2c), see [5,Equation (5.1)] or [6, Equation (2.6)] and that M (P n−1,n−1,c ) = TCPP(n, n, 2c), where TCPP(n, n, 2c) is the number of transpose-complementary plane partitions inside an (n, n, 2c)-box (see for example [3,Section 6]).Combining the above and the assertion (which is already proved above) we obtain immediately the numerical connection between SPPs and TCPPs stated in (1.5).
Finally we present another proof using non-intersecting lattice paths.We regard π as a lozenge tiling as above and draw n lattice paths in the following way.Each path ends at the top right boundary of the hexagon, uses the steps or and the i-th path from left has length n + 1 − i + c; see Figure 3 (left) for an example.By straightening the paths, we obtain non-intersecting lattice paths starting at A i = (2i, −i) and ending at E i = (n + 1 + i, c − i) with north-steps (0, 1) and east-steps (1, 0), see Figure 3 (right).For odd c, each family of paths has the same weight, namely 2 n .Hence the weighted enumeration is therefore by the Lindström-Gessel-Viennot Theorem equal to For even c we see that an anti-diagonal entry π n+1−j,j = c corresponds to the j-th path starting with an east-step.Define the points B 0 i = (2i + 1, −i) and B 1 i = (2i, −i + 1) which are reached from A i by an east-step or a north-step respectively.By deleting the first step of each path, we obtain a family of non-intersecting lattice paths starting from either B 0 i or B 1 i , where the weight is given by 2 to the power of times we start at B 1 i .Using again the Lindström-Gessel-Viennot Theorem we obtain for the weighted enumeration where we used the multilinearity of the determinant in the last step.Both determinants could be evaluated by guessing the corresponding LU decomposition and using the Pfaff-Saalschütz-summation formula, see for example [17, (2.3.1.3);Appendix (III.2)];we omit

Acknowledgement
The author thanks Ilse Fischer and Christian Krattenthaler for helpful discussions.We conjecture the following explicit formulas for qtcpp 2 (a, c) for 1 ≤ a ≤ 6.

Figure 1 .
Figure 1.A plane partition contained in a (3, 4, 4)-box on the left, its graphical representation as stacks of unit cubes (middle) and the associated lozenge tiling (right).

Figure 2 .
Figure 2. The labels of the corners of an (a, b, c)-box (middle), the corners where a copy of π is placed for self-complementary plane partitions (left), and the corners where copies of π are placed for quarter complementary plane partitions (right).The colour and lengths of the arrows indicate the orientation of the copies.

Conjecture 4 . 1 .
Let us denote by qspp(a, c) the number of quasi symmetric plane partitions inside an (a, a, c)-box.Then, qspp(a, c − a) = c c+a−1 2a−1 p a (c) a is even,

Appendix A .
Data from the computer experiments A.1.Quasi symmetric plane partitions.The values for the number qspp(a, c) of quasi symmetric plane partitions inside an (a, a, c)-box with a ≤ 6 and c ≤ 10 are shown in the next table.